Finite Group DatabasebetaFinite Group Databasebeta| order | 5832 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting | C4×#10624749 ⋊ C2 |
| Sylow | #10624749 ⋊ D4 |
1354 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 1354 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 4 generators on 31 points:
(2,4)(6,10)(7,13)(8,9)(11,20)(12,23)(14,25)(15,17)(16,19)(18,24)(21,27)(22,30)(26,28)(29,31);(1,2,3,4)(5,17,25,9,7,13,8,14,15)(6,26,31,12,11,24,16,22,21)(10,27,30,19,18,20,23,29,28);(7,14,17)(13,15,25)(18,27,29)(21,24,31);(5,6,10)(7,11,18)(8,16,23)(9,12,19)(13,24,20)(14,22,29)(15,21,28)(17,26,27)(25,31,30)Polycyclic presentation:
2^3,3^6|a^2=c,f^3=i,b^a=bc,d^b=d^2,e^b=e^2,e^d=eh,f^b=f^2i^2,f^e=fg,g^b=gi^2,g^d=gi,g^f=gi,h^f=hi^2,i^b=i^2141 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#13091120, #72276036, #72276091 |
| maximal normal subgroup of | D4×#13091120, #72276036, #72276091 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6^2*F.9^2);;
gap> SetConjugate(coll, 6, 5, F.6*F.7);;
gap> SetConjugate(coll, 7, 2, F.7*F.9^2);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.9^2);;
gap> SetConjugate(coll, 9, 2, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);